Download The Two Triangles: What Did Wicksell and Keynes Know about

The Two Triangles: What Did Wicksell and Keynes
Know about Macroeconomics that Modern
Economists do not (consider)?∗
Roberto Tamborini
University of Trento
[email protected]
Hans-Michael Trautwein
University of Oldenburg
[email protected]
Ronny Mazzocchi
University of Siena
[email protected]
This version: September 2009
Abstract
The current consensus in macroeconomics, as represented by the
New Neoclassical Synthesis, is to work within frameworks that combine
intertemporal optimization, imperfect competition and sticky prices.
We contrast this NNS triangle with a model in the spirit of Wicksell and Keynes that sets the focus on interest-rate misalignments as
problems of intertemporal coordination of consumption and production
plans in imperfect capital markets. We show that, with minimal deviations from the standard perfect competition model, a model structure
can be derived that looks similar to the NNS triangle, but yields substantially different conclusions with regard to the dynamics of inflation
and output gaps and to the design of the appropriate rule for monetary
policy.
JEL classification: E20, E31, E32, E52, D84
∗
We wish to thank Robert M. Solow, Peter Spahn and the CNRS at Université de Nice
- Sophia Antipolis for helpful comments and discussions.
1
Introduction
If anyone is to blame for the awkwardness of the title of this paper, it
should be Olivier Blanchard. Looking back on the 20th Century, Blanchard
(2000) raised the question: ”What do we know about macroeconomics that
Wicksell and Fisher did not?” His answer was: ”. . . a lot”, stressing that
much progress has been made in modelling short-run fluctuations of aggregate output as effects of ”imperfections”, i.e. as results of ”deviations from
the standard perfect competition model”1 . In another prominent review,
Blanchard (1997, p. 290) observed that almost all macroeconomists now
work within a framework that combines three ingredients: intertemporal
optimization, imperfect competition and sticky prices. This combination
characterizes the current consensus view that is generally labelled as the
New Neoclassical Synthesis (NNS), or the NNS triangle.
The NNS is epitomized by Michael Woodford’s Interest and Prices (2003b),
a book that explores the properties and welfare implications of monetary policy in the confines of the standard IS-AS-MP framework, where aggregate
demand (IS) is derived from the representative household’s intertemporal
utility maximization, and aggregate supply (AS) is expressed in terms of
a New Keynesian Phillips curve, based on imperfect competition and price
rigidities. The model is closed by a reaction function of monetary policy
(MP), typically a Taylor rule. Woodford (2003b, ch. 4) describes the model
as ”neo-Wicksellian”, since the IS relation nests the real interest rate associated with potential output and the resulting (possibly zero) inflation rate
the ”natural rate of interest” in Wicksell’s diction. Accounting for stochastic
changes of this rate that follow from shocks to technology and preferences,
the evolution of the economy over time is represented as a dynamic stochastic general equilibrium (DSGE) process. As a consequence of sticky prices,
any gap between the market real interest rate (the nominal rate minus expected inflation) and the natural rate generates gaps in output and inflation
with respect to their optimal values. These gaps, which entail welfare losses,
can be reduced by appropriately designed interest-feedback rules on which
the central bank operates.
This apparatus is clearly reminiscent of Knut Wicksell’s Interest and
Prices (1898). Referring to Wicksell and his followers Friedrich A. Hayek,
Erik Lindahl and Gunnar Myrdal, Woodford (2003b, p. 5) claims ”to resurrect a view that was influential among monetary economists prior to the
Keynesian revolution”. Moreover, by setting the focus on output effects of
false positions of the market rate of interest, his version of the NNS appears
to bring a common theme of Wicksellian and Keynesian theories back to
the forefront of modern macroeconomics.
However, Wicksell and his followers, as well as Keynes, did not work
1
For a similar history of 20th Century macroeconomics see Woodford (1999).
2
within the NNS triangle. They normally used the assumption of ”free competition”. The stickiness of prices and wages played, at most, a subordinate role in their explanations of macroeconomic fluctuations. Their key
”deviation from the standard perfect competition model” was an imperfection of the price mechanism in the capital market, not in the goods or
labour markets. Even though they differed in their conceptual frameworks,
both Wicksell and Keynes set the focus on the causes and consequences
of misalignments of the market rate of interest with the rate, at which
the plans of real investors (business firms) and financial investors (savers,
wealth-holders) would be fully coordinated2 . Intertemporal disequilibrium
mattered to Wicksell and Keynes, not because of quantity rationing, but
because, as long as the market real rate of interest is at the ”wrong level”,
the economy will be driven away from its intertemporal-equilibrium path in
a cumulative process of changes in prices and/or output at which capital
and goods markets clear 3 . They also shared the view that price and wage
flexibility would not per se eliminate the effects of wrong interest rates nor
induce their correction. Describing the ”individual experiment” of optimal
household planning, as in the NNS, was therefore not considered sufficient
to determine the time paths of output and inflation. The ”market experiment” i.e. the relative speeds and sizes of output and price adjustments,
when intertemporal equilibrium is disturbed by interest-rate misalignments
requires explicit analysis of the resulting shifts in the budget constraints4 .
Therefore, by way of contrast with the NNS triangle of intertemporal optimization, imperfect competition and sticky prices, a ”Wicksell-Keynes (WK)
triangle” can be described by the key words ”intertemporal coordination”,
2
In Wicksell’s bank-centred credit economy, it is the bank’s lack of information about
the full-employment-equilibrium, or ”natural”, rate of interest that causes the misalignments of the market rate. In Keynes’s monetary economy, it is the liquidity preference of
wealth holders in view of uncertainty about returns to investments that drives a wedge
between the two rates. While Keynes had made use of Wicksells natural-rate concept
in his Treatise (1930), he criticized it in the General Theory (1936, p. 183, 242-44) for
its underlying loanable-funds view, but retained the reference to a benchmark rate that
would be compatible with full employment. His shift towards liquidity-preference theory
has led Leijonhufvud (1981) to argue that Keynes lost his ”Wicksell Connection” when
writing the General Theory. However, in the wider sense of coordination failures of the
interest-rate mechanism, Leijonhufvud draws a dividing line between Wicksell and Keynes
on the one hand, and the Keynesians of the old Neoclassical Synthesis whose explanations
of macroeconomic fluctuations revolve around price and wage rigidities, on the other. Here
we follow the same line with regard to the New Keynesians in the NNS.
3
Wicksell (1898) restricted his analysis of the cumulative process to changes in the
price level, whereas Keynes (1936) discussed cumulative changes of output, though neither
of them denied that these processes involve changes in both output and prices. Lindahl
(1930) and Myrdal (1931) analysed combinations of both, whereas Hayek (1931) described
a change in the structures, not the levels, of prices and production. As we shall see, changes
in all these dimensions coexist.
4
For the distinction between individual and market experiments see that other book
with interest and prices in its title: Patinkin (1989, ch.I:4).
3
”imperfect capital markets” and ”wrong interest rates”.
Do the differences between the two triangles matter? Is there anything
we can still learn form the WK triangle which is not in the NNS triangle?
The standard answer is that macroeconomics, in the days of Wicksell and
Keynes, lacked ”conceptual rigour” in two regards. Fluctuations of output
and price levels were not fully modelled in terms of optimizing behaviour,
and the ”imperfections” that explain them were not well understood (e.g.,
Blanchard, 2000, p. 1385-88; Woodford, 2003b, p. 5-6). The NNS is believed to have overcome these shortcomings by integrating price-setting behaviour into a DSGE framework. Woodford’s insistence that the key results
of Wicksellian theory can be replicated in DSGE models (Woodford, 2003b;
Woodford, 2006, p. 196-97) suggests that the differences between the two
triangles do not matter substantially, so that the older triangle could safely
be left to historians.
The purpose of our paper is to challenge this view. In a more recent
survey, Blanchard (2008) discusses several points where the standard NNS
model should be improved to include a larger array of phenomena and provide better explanations. In this perspective, our point is that the boundaries of the NNS triangle hamper progress in dealing with relevant issues
raised by the older theories. In particular, ignoring saving-investment imbalances seriously limits the explanatory scope of the NNS. Not only are
these phenomena a logical implication of any theory based on the distinction between the market real interest rate and the Wicksellian natural rate.
It can also be argued that informational imperfections in the capital markets
and the consequent intertemporal imbalances play a major role in processes
that generate macroeconomic instability, as testified by the global financial
crisis that has recently developed (e.g. Borio and Lowe, 2002; Leijonhufvud,
2008)5 .
In the following, we therefore propose a modeling framework that reconstructs, with some rigour, the WK triangle. Our model is meant to be
authentic, in the sense that it is made of components that were available
at the times of Wicksell and Keynes, if not used or even devised by them.
Even though we will provide textual evidence, our rendition is not, however, meant to be ”true” to these writers in every detail. At the same time,
we have devised the model to be closely comparable to the standard threeequations model of the NNS by means of current modeling techniques. In
5
Saving-investment imbalances are ignored even in the NNS model which include fixed
capital and investment (e.g. Woodford, 2003b, ch. 5; Casares and McCallum, 2000). One
strand of earlier New Keynesian literature had set the focus on imperfect information and
capital market failures (see, e.g., Greenwald and Stiglitz, 1987; Bernanke and Gertler,
1990). To the extent that its results have been incorporated into the NNS, they are not
connected with the possibility of saving-investment imbalances and their intertemporal
implications. The same applies to NNS extensions that include imperfect information and
investment dynamics (see, e.g., Woodford, 2003b, ch. 5; Woodford, 2003a; Mankiw and
Reis, 2003). We comment on these extensions below.
4
this way, we highlight what Wicksell and Keynes knew about macroeconomics that might still be worth knowing, even though modern economists
tend to exclude it from their consensus view. We demonstrate the explanatory potential of a synthesis of Wicksellian and Keynesian ideas that differs
substantially from the Neoclassical Synthesis, Old and New.
The rest of the paper is organized in three parts. Section [2] provides
a brief account of the standard NNS model (Woodford, 2003b, ch.4) that
serves to draw attention to critical limitations of the new synthesis. We
discuss its effective reduction to the intratemporal coordination of aggregate
demand and supply, and its specific assumptions about market structures
that must be inefficient enough to create the relevant output gaps, while
excluding major logical implications of interest-rate misalignments, namely
the causes and consequences of a mismatch of intertemporal consumption
and production plans.
Section [3] introduces the model of an economy in the confines of the WK
triangle. Prices are flexible, agents have forward-looking expectations, the
economy has a nominal unit of account, and physical capital is represented
by interest-bearing bonds traded in the capital market. The central bank is
the only policymaker, taking control of the economy’s nominal interest rate
by trading bonds. Information is imperfect in the sense that no one in the
market can directly observe the natural rate of interest, i.e. the real rate
that would equate optimal saving and investment. This lack of information
is sufficient to generate ”interest-rate gaps” and intertemporal coordination
failures between saving and investment.
The thrust of the model is that saving-investment imbalances, by way of
forward-looking agents’ allocations, generate an intertemporal spillover effect, in that they transmit the effects of present interest-rate gaps to present
and future output and inflation. As a result, these variables diverge from
their intertemporal equilibrium path, in a way that considerably modifies
the dynamic properties of the system with respect to the standard NNS
framework where this effect is not present. First, output and inflation gaps
display the autocorrelated dynamic structure that is typically observed in
the data, with no addition of further frictions and imperfections. Second,
it is shown that the system displays three distinctive WK features. Given
an interest-rate gap, the time paths of prices and output evolve as ”cumulative processes”, which are the result of the interplay of ”deep parameters”
rather than exogenous factors. The system will return to its intertemporalequilibrium path only when the interest-rate gap is closed and the resulting
saving-investment imbalance is corrected. Hence, price (or wage) stickiness
is not the core problem, price (and wage) flexibility not the general solution.
In section [4] we focus on the formation of inflation expectations and on
their implications for the system’s stability as well as monetary policy. Since
the model allows us to analyse out-of-equilibrium dynamics that are beyond
the scope of the NNS triangle, we can show how short-run rational expecta5
tions (the standard assumption in the NNS world) amplify price and output
dynamics and tend to destabilize the system, as argued by both Wicksell and
Keynes. This result provides a consistent foundation for those two ideas of
Wicksell that have been transplanted into the NNS theory of monetary policy namely, the idea that stability can be preserved by a ”nominal anchor”
in terms of a stable price level (or inflation rate) in which agents have reason
to believe, and the idea that, to this effect, the nominal interest rate should
be raised (lowered) as soon as the rate of inflation increases (decreases).
In fact, we show that a simple ”Wicksellian rule” for monetary policy that
follows this prescription without any other informational requirements may
display remarkable out-of-equilibrium virtues of system stabilization. We
also demonstrate that, based on the theoretical framework of the WK triangle, this kind of rule has dynamic properties that stand in contrast with
the NNS literature. In particular, stability requires that the inflation coefficient of the rule should have an upper bound. This implies that the so-called
Taylor principle, which requires an overproportional interest-rate reaction
to the inflation gap (e.g. Woodford, 2003b, ch. 3), is not necessary and may
well be conducive to instability. Central banks may thus face a trade-off,
not between inflation and output control, but between ”small gaps” and
”smooth paths” in the adjustment process. In the final section we draw the
conclusions and point out in which directions the Wicksell-Keynes triangle
may be further explored.
2
The Limitations of the New Neoclassical Synthesis
In this section we sketch the basic model of the NNS in its most influential
version, as presented by Woodford (2003b), and discuss its analytical limitations to set the stage for the subsequent exposition of our alternative WK
model.
2.1
The core of the NNS model
The NNS is based on a system of three equations that determines the shortrun dynamics of output, inflation and interest rates. Confining his version
to the study of ”small fluctuations around a deterministic steady state”,
Woodford (2003b, p. 243-47) describes it as a log-linear approximation of
the conditions for intertemporal general equilibrium.
The first equation describes aggregate demand and resembles the IS
curve of the old Neoclassical Synthesis, insofar as it displays a negative
relation between current output and the current (real) interest rate. It
is, however, entirely based on the representative household’s consumptionsaving decision and obtained by log-linearizing the first-order condition of
6
maximizing utility over time:
xt = Et xt+1 − σ(it − Et πt+1 − rt∗ )
(2.1)
As intertemporal optimization is formulated in terms of deviations from the
steady state, xt denotes the gap between actual output and the ”natural
rate of output”, σ is the (constant) intertemporal elasticity of substitution
of aggregate spending, it is the nominal interest rate Et πt+1 is the rational
expectation of the inflation rate conditional on information available at time
t. Finally, rt∗ is the ”natural rate of interest”. This terminology refers to the
fact that rt∗ is the value of the market real interest rate, it − Et πt+1 , that
is consistent with the steady state value of the output gap, which is zero.
As will be seen, x∗ = 0 is also consistent with zero inflation. By contrast,
deviations of the market real interest rate from rt∗ trigger non-zero output
and inflation gaps. In other words, the natural rate of interest is ”just the
real rate of interest required to keep aggregate demand equal at all times
to the natural rate of output” (2003b, p.248). The ”natural rate of output”
refers to a virtual equilibrium with price flexibility, i.e. to the output ”one
would have if prices and wages were not in fact sticky” (2003b, p.9).
The second equation describes aggregate supply by relating the output
gap to inflation. The AS function is labelled ”New Keynesian Phillips curve”
(though little reference is made to the labour market), since actual inflation
is proportional to expected inflation and the output gap,
πt = βEt πt+1 + κxt
(2.2)
with β denoting a discount factor, and κ a rigidity parameter. In order to
introduce nominal rigidities, Woodford (2003b, ch.2) makes the assumption
that consumer demand is met by the supply of a variety of differentiated, imperfectly substitutable goods. This allows firms to set prices in monopolistic
competition, following the Dixit-Stiglitz (1977) model. Imperfect competition does not per se imply price rigidities, so it is assumed that prices are set
in a staggered fashion. In the case of interest-rate shocks (changes in it or
rt∗ in equation (2.1)), a significant fraction of firms will maximize profits by
varying their output rather than prices. The value of the rigidity parameter
κ rises with the strategic complementarity of price-setting decisions, such
that the output effects of shocks can become large and persistent.
The third equation describes the feedback of the interest rate to changes
in inflation and output gaps, assuming that the central bank controls the
representative nominal interest rate. Woodford (2003b, p. 245) writes the
MP function in terms of a Taylor rule:
it = i∗t + γπ (πt − π ∗ ) + γx (xt − x∗ )
(2.3)
where i∗ is an intercept term that correspond to the nominal value of the
natural interest rate, r∗ + π ∗ , or NAIRI (non-accelerating-inflation rate of
7
interest). The weight factors γπ and γx describe the intensity of the interestrate reactions to deviations of actual inflation and the output gap from their
target values. The target for the output gap is defined as the steady-state
value consistent with the inflation target. Using (2.2), this implies that
∗
∗
∗
x∗ = 1−β
κ π , and ensures that it = it whenever the inflation target π is
achieved. The MP function closes the model, permitting the determination
of the endogenous variables it ,xt and πt .
The IS-AS-MP framework can thus be characterized as a synthesis of
standard Neoclassical, Wicksellian and Keynesian ideas. The failure of the
system to converge automatically on its natural rates of output and interest
is ascribed to nominal rigidities outside the capital market, as in the ”classically” neoclassical approaches of Cassel (1918) and Pigou (1933), or in
the old Neoclassical Synthesis á la Modigliani (1944). As in Keynes (1936),
output adjustments accord with aggregate demand and precede, or even prevent, price adjustments, such that the latter do not automatically restore
the optimal position of the system. As in parts of the Wicksellian literature,
the ”neutrality of money”6 is not an automatic outcome of market processes,
but requires a specific political strategy of interest-rate feedbacks to changes
in the price level and/or output.
2.2
Some implications of the model
In the following we draw attention to assumptions in the basic ”neo-Wicksellian” model, and indeed in much of the NNS literature in general, that limit
the scope of the analysis of output gaps. Here we discuss only those that
are most important for the contrast with the alternative Wicksell-Keynes
model presented in the following section7 .
It should be noted that the standard formulation of the IS relation (in
equation (2.1)) implies that aggregate demand consists just of consumption. As Woodford (2003b, p. 242) points out, the model ”abstracts from
the effects of variations in private spending (including those classified as investment expenditure in the national income accounts) upon the economy’s
productive capacity”, so the model should be interpreted ”as if all forms of
private expenditure. . . were like nondurable consumer purchases”8 . Thus rt∗
6
It may be argued that money is not a quantitatively well-defined concept, neither in
Woodford (2003b) nor in Wicksell (1898, 1898a); see, e.g., Laidler (2006). Furthermore,
Woodford seems to avoid to refer to the concept of neutrality of money, while Wicksell
confined real effects to the small print. Yet, what is at stake here is the argument that
monetary policy has the power to change interest rates such that both the price level and
the structure of prices will be affected. Thus it can produce (or avoid) substantial effects
of monetary policy on output and income distribution, as argued, for example, by Lindahl
(1930), Hayek (1931) and Myrdal (1931).
7
Other problematic assumptions, especially those pertaining to the monetary foundations of Woodfords model, are examined in Boianovsky and Trautwein (2006). Some of
their implications will be discussed in sections [4] and [5] below.
8
Woodford (2003b, ch. 5) extends the basic model to include fixed capital and the
8
is best interpreted as the representative household’s rate of time preference.
If the market real interest rate rises, and if there is no corresponding shift
in rt∗ , current consumption falls in favour of increased future consumption.
The opposite occurs if rt∗ rises and is not matched by shifts in the market
real interest rate.
The simplifying exclusion of (net) investment has two implications. First,
changes in consumption plans translate themselves into one-to-one changes
in aggregate demand at each date. Second, the intertemporal coordination
problem between future consumption (saving) and future production (investment), which is the key problem to be solved by the interest rate in
general equilibrium theory, vanishes. The system is effectively reduced to
the intratemporal coordination of current aggregate demand and supply in
each period, essentially (to be) accomplished by the system of spot prices
for goods.
Woodford (2003b, p. 71 and elsewhere) assumes that ”markets must
clear at all dates”. It should be noted that the NNS uses a deviation from
the ”standard perfect competition model” contained in the AS function, and
not in the IS relation to provide a mechanism that coordinates aggregate
demand and supply continuously. The key ”imperfection” is monopolistic
competition, which in the macroeconomic literature is nowadays treated as
a standard ”source of inefficiency” (cf. Cooper, 2004). Even so this market
structure allows firms to determine supply and set prices while consumers
are always on their demand curve. In the spirit of Blanchard and Kiyotaki
(1987), this is frequently interpreted as a resurrection of the Keynesian message that aggregate output is governed by aggregate demand, at least in the
short run. However, if it is not simply postulated that aggregate demand
always matches aggregate supply, further conditions must be satisfied to ensure that state. Thus it must be assumed that the households in the system
own the firms, that they receive the firms’ profits as part of their income, and
that there are no distribution effects of output gaps and inflation that could
feed back onto the latter. All this is implicit in the conventional assumption
of the representative household. Finally, the model is based on the premise
that households and firms have rational expectations of future output gaps
and inflation. In this setting, the optimal plans are self-fulfilling, at least in
the absence of shocks. Output gaps and inflation have no influence on the
speed and extent to which information is incurred, processed and used for
predictions.
Imperfect competition implies price-setting behaviour, which helps to introduce nominal rigidities. The latter do not automatically follow from the
model of monopolistic competition. To define the natural rate of output,
Woodford (2003b) actually uses the strong assumption of a monopolistieffects of the related investment dynamics, but the possibility of unplanned saving is
excluded by assumption.
9
cally competitive system with flexible prices, in which output growth would
not substantially differ from that of a perfectly competitive system. This
is the fictitious NNS benchmark for assessing the welfare losses that accrue from sticky prices. The existence of nominal rigidities in the goods
markets is, in turn, crucial for generating suboptimal equilibria in the NNS
framework. They are introduced with reference to menu costs that make
profit-maximizing firms choose between price and output adjustments in response to disturbances that affect marginal costs (Woodford, 2003b, ch. 3).
However, these menu costs and other components of the rigidity parameter
in equation (2.2) are exogenously given.
The NNS triangle is thus a narrowly confined space, precariously based
on specific assumptions, by which market structures are postulated that
must meet conflicting requirements. On the one hand, they must be inefficient enough to generate the output gaps that form the control problem for
the social planner, as embodied in (2.2), the Taylor rule for monetary policy.
On the other hand, they must rule out all the saving-investment imbalances
that were at the heart of the distinction between market rates and natural
rates of interest in earlier macroeconomics. If there are financial markets
constituted by independent borrowers and lenders, the consequence of the
market real interest rate on loans being higher (lower) than the natural rate
is that households wish to save more (less), whereas firms wish to invest
less (more). Neither side of the market can achieve intertemporal equilibrium, and the constraints that follow from the interest-rate gap need to be
examined as out-of-equilibrium dynamics. This problem was in the focus of
Wicksellian and Keynesian economics in the early 20th Century, whereas it
is not contemplated even in the models that endogenize investment into the
NNS framework (such as Casares and McCallum, 2000; Woodford, 2003b,
ch. 5). While the distortionary effects of sticky prices are the raison d’être
of monetary policy in the NNS, Wicksell (1898) argued that interest rates
should be brought under policy control, not because there is a lack of price
flexibility in the goods markets, but because misalignments of interest rates
may force prices to move out of equilibrium. Keynes (1936, ch. 19), too,
emphasized that there is no automatism, by which price and wage adjustments in goods and labour markets could balance the effects of a misaligned
market rate of interest; he stressed that price flexibility would actually tend
to make things worse. In the following, we will examine these issues, which
cannot be captured in the confines of the NNS triangle, while they can - with
some rigour - be explored in the framework of the Wicksell-Keynes triangle.
3
Modelling the Wicksell-Keynes Triangle
In the following we present the basic model of an economy in the WK triangle. As it is conceived to highlight the differences between the macroe-
10
conomics of saving-investment imbalances and the NNS triangle, the model
leaves the internal differences between Wicksell and Keynes in the background. It is Wicksellian in that it retains the assumption of a natural
interest rate as gravitation centre of the economy, with no consideration of
liquidity preference. It is Keynesian in that real income adjustments occur
as a direct implication of intertemporal disequilibrium, notwithstanding the
flexibility of prices. Apart from being based on insights of Wicksell (1898)
and Keynes (1930, 1936), the model includes features that were emphasized by other authors in the Wicksellian tradition, such as Lindahl (1930),
Myrdal (1931) and Lundberg (1930, 1937).
3.1
The basic setup
The economy consist of three competitive markets (for labour, capital and
output) and rational forward-looking agents. All exchanges take place in
terms of a general unit of account of value P1t , where Pt is the general
price level. It is assumed that technology and consumer preferences take
the specific forms, respectively, of a Cobb-Douglas production function and
a logarithmic utility function. These specific restrictions are not necessary,
but they are useful to obtain a manageable closed-form solution to the model,
and to make it comparable to standard modern models9 .
Hence, aggregate output Yt , which consists of a homogenous good that
can be consumed or used as input in the production process, is given by
Yt = Kta Lbt where a + b = 1
(3.1)
where Kt is the capital stock available at time t and Lt is the current input
of labour. The share of output that is transformed into capital at time t
becomes operative in the production process only in the next period, t + 1.
Capital is fully depreciated within one period, so that
Kt+1 = It0
(3.2)
where It0 = It + Kt is gross investment including capital replacement and It
is net investment.
9
As to the ”authenticity” of the model, the use of these ”classically neoclassical” assumptions may be surprising, in particular to those who connect the ideas of Keynes and
other post-Wicksellians to a fundamental critique of the neoclassical theory of income
distribution. It should be noted, however, that Wicksell (1901, p. 128) was one of the
pioneers of the concept of the Cobb-Douglas production function, using it to show that the
exhaustion theorem (according to which aggregate output is exactly absorbed by aggregate
income, if the prices of the production factors accord with their marginal productivity) is
valid only if returns to scale are constant. That assumption is restrictive, but certainly no
bigger ”as if” than the NNS assumption of an insignificant wedge between total output
under imperfect competition with flexible prices and its perfect-competition counterpart
(see above, section [2.2]).
11
Firms are price takers and seek to maximize their expected stream of
profits, given (3.1) and their costs in terms of the income-distribution constraint:
!
∞
X
(3.3)
Yt+s − wt+s Lt+s − Rt+s Kt+s
Et
s=0
where for each period t, wt is the real wage rate and Rt ≡ 1 + rt is the real
gross return purchased at time t − 1. For each period t, firms’ programmes
consist of the choice labour L∗t for current production, and the capital stock
∗
Kt+1
for the next one.
In the labour market, workers and firms bargain over a real wage before
production takes place, but labour contracts are in nominal terms, Wt . The
nominal wage rate is obtained by way of indexation of the negotiated real
rate wt∗ to the expected price level Pte , where e denotes an expectational
variable (whose formation will be discussed below). Hence, denoting with πt
the one-period inflation rate, the nominal wage rate results Wt = wt∗ Pt−1 (1+
πte . Subsequently, firms choose L∗t for production, observing the actual real
wage rate given bye the nominal rate deflated by the actual price level, wt ≡
Wt
∗ 1+πt 10
Pt or wt = wt 1+πt .
Firms can raise funds to invest in the capital stock by selling one-period
bonds. By analogy with physical capital, they are time-indexed by maturity.
Hence Bt+1 (in real terms) denotes bonds issued in period t that bear a
nominal interest rate it with maturity in t + 1. Likewise, the real rate of
t−1
return to capital that firms pay to bond-holders in t is Rt ≡ 1+i
1+πt , whereas
1+it
. As a result, L∗t and
the rate relevant to investment in t is Rt+1 ≡ 1+π
e
t+1
∗
Kt+1
should satisfy the first order conditions of (3.3):
a
Kt
(1 − a)
= wt
(3.4)
L∗t
∗ 1−a
Lt
a
= Rt+1
(3.5)
∗
Kt+1
Households hold claims to the capital stock and supply their whole labour
force L inelastically, which is normalized to unity11 . They choose a consumption plan (Ct+s ; s = 0, 1, . . .) in order to maximize their lifetime expected
utility
#
"∞
X
Et
Θ−s ln Ct+s
(3.6)
s=0
10
This representation of the labour market is akin to Keynes (1936, ch. 19).
Whether the equilibrium employment is equal to, or less than, the total labour force,
or how wages are determined, is immaterial here, or as implicit as the labour market is
in Woodford’s (2003b) version of the NNS. In analogy with the NNS one may allow for
some ”natural rate of unemployment” that creates a reserve capacity of labour. On the
implications of this assumption see our discussion of the WK model in section [4] below.
11
12
given the constant rate of time preference θ > 0, Θ ≡ (1 + θ), and the
dynamic budget constraint:
Bt+1 = Ht + Rt Bt − Ct
(3.7)
Here Ht = wt Lt is labour income, Bt is the real stock of bonds representative installed capital, and the operator Et (·) denotes forward-looking expectations, conditional upon information available at time t. Consequently
Bt+1 − Bt = St is net saving, expressed as real purchases of bonds. Given
their utility function (as in (3.6)), the households’ optimal consumption as
of time t is expressed as
Ct+1
Ct = Et
Θ
(3.8)
Rt+1
In order to insulate the macroeconomic effects of interest-rate gaps, the
analysis should start at a point in which the economy is in intertemporal
general equilibrium (IGE). It is easily seen that equations (3.1), (3.4), (3.5)
e
and (3.8), given L∗ = 1 and πt+1
= πte = πt , yield the steady-state solution,
where the real return to capital is Rt+1 = Θ = R∗ , so that for all t, Ct = C ∗ ,
B ∗ = K ∗ and Y ∗ = H ∗ +R∗ K ∗ . Note that St0 = It0 = K ∗ , and once account is
taken of capital replacement, net investment and saving are nil12 . Hence R∗
is also the natural interest rate that equates optimal saving and investment.
What remains to be determined is the price level and the steady-state inflation rate. As is well-known, a competitive general equilibrium model like
the present one leaves the price level and its changes undetermined. The
usual practice has long been to introduce a special ”monetary equation”
that pins down the price level corresponding to the general-equilibrium output level. This is actually the critical point where Wicksell departed from
the classical tradition of the quantity monetary equation and introduced his
interest-rate theory of inflation. Once the nominal interest rate is in place,
the intertemporal general equilibrium characterized above also implies that
e ) should hold for all t. This relationship is also known
(1 + it ) = R∗ (1 + πt+1
as ”Fisher equation”, and is in fact plugged into the NNS model (equation
(2.1)). Yet, from a Wicksellian point of view, the Fisher equation presents
two problems. First, if it is employed to pin down the nominal interest rate,
it still leaves the price level determination unresolved, and the expected inflation term on the right-hand side is left hanging on its own bootstraps.
Second, if expected inflation rate is to be determined by postulating an
exogenous nominal interest rate, it should be high as long as the nominal
interest rate is high relative to the natural rate, and vice versa - in contradiction with the conclusions of Wicksell, the NNS and common sense (but
argued by McCallum, 1986). The Fisher equation holds in intertemporal
12
Abstracting from techinal progress or technical shocks, this is in fact a Sidrausky-type
steady state, where the key allocational variable in the capital market is just the rate of
intertemporal consumer preferences, Θ.
13
equilibrium, but in the construction of a consistent interest-rate theory of
the general price level, it cannot be taken to hold continuosly. Such a theory
requires three building blocks: (i) the market determination of the nominal interest rate, (ii) the determination of inflation expectations, and (iii)
the interaction between the two. The whole should be analysed in terms of
an intertemporal disequilibrium process to check whether it eventually converges to the Fisher equality and how much inflation will be generated in
the process. This was in fact the research agend of Wicksell and Lindahl.
Since this is the most critical nexus in the whole construction, we wish
to proceed step by step from the simplest hypotheses concerning the determination of the nominal interest-rate and of inflation expectations to
alternative hypotheses drawing on Wicksell and the subsequent literature.
In analogy with Wicksell (1898), Lindahl (1930) and Keynes (1930; 1936),
we begin with a simplified monetary system that consist of a central bank,
representative the system of bank loans and deposits as a whole. In a Keynesian twist of this assumption, the central bank seeks to gain control over
the nominal interest rate it by buying and selling bonds in the open market.
Pegging the nominal rate, the central bank is ready to create or retire base
money (the counterpart of bonds) to the extent that is necessary to clear the
market13 . All agents form their expectations of the inflation rate, consider
their objective functions and constraints, and make their plans accordingly.
At this stage, we simply posit that all agents believe in a time-invariant
e
”normal” rate, πt+1
= π e = π ∗ 14 . Finally, exchanges occur in all markets,
and [Yt , πt ] are realized.
3.2
Three-gap analysis
To begin with, we examine the implications of a gap arising between the market real interest rate and the natural rate with respect to the intertemporal
general equilibrium (IGE) characterized above. Given π ∗ , a gap between
Rt+1 and R∗ can arise owing to a nominal cause, where it is set or changed
inconsistently with R∗ (1 + π ∗ ), or to a real cause, where a change in R∗
is not matched by a change in it . Which of the two causes originates the
gap is immaterial here. The key issue is how the system behaves as long as
13
Admittedly, this representation can be viewed as a short-cut with respect to the fullblown analysis of the banking sector by Wicksell. However, the key point remains, as the
central banks’ willingness to create (destroy) reserves at the given nominal interest rate
corresponds to the aggregate effects of private banks lending.
14
The concept of the ”normal” value of a variable was widely used as point of reference
expectations by Wicksell, Lindahl, Keynes and pre-Lucasian economists in general. It
normally referred to the long-run average value observed for the variable in question,
expected to prevail in future states of rest of the system. For simplicity, this information
about inflation is taken here as the pre-determined (possibly zero) value π ∗ . If it also
results to be the steady-state solution for inflation, then π ∗ is also the ”long-run” rational
expectation of the inflation rate.
14
Rt+1 6= R∗ . What Wicksell and Keynes had in mind is, in modern terminology, ”trading at false price” (which should not be confused with quantity
rationing). By pegging it , the central bank keeps clearing the bond market:
as long as Rt+1 > R∗ , it meets excess saving with extra-sales of bonds; the
converse applies in the case of Rt+1 < R∗ . The crucial consequence of this
can be expressed in the following proposition (see also the Appendix):
Proposition 1 Given Rt+1 6= R∗ , although the bond market clears, the
ensuing levels of saving and investment are not consistent with the clearing
of the goods market, neither in t nor in the subsequent periods, at the IGE
values of output and and inflation.
The proof of this proposition is a well-known implication of Walras Law,
which yields the constellations in the goods market that are associated with
Rt+1 6= R∗ in the bond market:
Bond Market (at time t)
Output Market (at time t)
Output Market (at time t + 1)
Rt+1 > R∗
exc. supply
exc. demand
Rt+1 < R∗
exc. demand
exc. supply
Consider the typical Wicksellian case Rt+1 < R∗15 . In this case, firms are
allowed to invest more (adding physical capital to their net worth) than
households are actually ready to save (adding more bonds to their wealth).
The consequence is excess demand in period t due to net investment and
excess supply in period t + 1, corresponding to a production capacity (due
to net investment in t) that is not matched by planned consumption (low
saving in t). If we abide with the principle that ”markets always clear”,
then we should explain how can these intra- and inter-temporal inconsistencies among plans can be brought into equilibirum. The solution lies in the
following proposition (proof in Appendix):
Proposition 2 Given Rt+1 6= R∗ in any period t, there exists one single
sequence of realizations of output and inflation in t and owards that clears
the goods market.
In consideration of Proposition 1, such a sequence cannot be identical to
the one that would obtain in intertemporal equilibrium, with Rt+1 = R∗ .
In fact, if we follow Woodford’s procedure of relating actual output and
inflation at each point time, Yt , πt , to their respective IGE value, Y ∗ , π ∗ ,
we obtain the following expressions in terms of ”gaps” (see Appendix):
15
− 1
− a
1−a
1−a
Ŷt = R̂t+1
, Ŷt+1 = R̂t+1
(3.9)
a
a
1−a
1−a
, Π̂t+1 = Ŷt+1
Π̂t = Ŷt
(3.10)
Keynes was more concerned with the opposite case, but the mechanism is the same.
15
1+πt
where Ŷt ≡ YYt∗ , R̂t+1 ≡ RRt+1
and Πt ≡ 1+π
∗
e.
To understand these results consider our previous example with Rt+1 <
R∗ , i.e. the case of excess investment, with the central bank buying extra bonds in the market. This case makes households in period t reckon
a real value of wealth (bonds) smaller than the value of capital the bonds
are supposed to represent. The interest-rate gap affects the accounting of
real resources in the economy, and the economy needs a correction of the
intertemporal resource distribution. To this effect, output (real incomes
accruing to households) should be higher along the consumption path of
households. Correspondingly, in order to induce profit-maximizing firms to
increase output with respect to Y ∗ , the inflation rate, too, should be (unexpectedly) higher than the normal rate embedded in nominal wage contracts.
Note that unexpected inflation is an integral part of the process, in the
sense that, as long as there exist interest-rate gaps, and hence output gaps,
the economy must be off whatever inflation path expected by agents. As
a matter of logic implied by the rational-expectations hypothesis, agents
form inflation expectations in consistence with their plans. However, as
these plans are frustrated in the goods market, the related expectations of
inflation will be falsified.
3.3
A log-linear version
To facilitate comparison with the NNS, we now present a log-linear version
of the previous model. Let us consider the relationship between the market
real interest rate and the natural rate in any period t, and let us begin with
the IS function (3.9). Since the output gaps in t and in subsequent periods
share the common factor RRt+1
∗ , they can also be expressed in a single reduced
form that, in its log version with Woodford’s notation, yields16 :
ŷt+1 = ρŷt − α(it − π ∗ − r)
(3.11)
There is a clear analogy with the IS (equation (2.1)) in the NNS model, but
there are substantial differences, too. Equation (3.11) describes output dynamics off the IS schedule that corresponds to the IGE in the way explained
above. Due to their intertemporal ”feed-forward effect” (which is not captured by the NNS model), interest-rate gaps generate time series of output
gaps that display (a) dependence on the lagged value of interest-rate gaps,
and (b) some degree of (spurious) serial correlation or ”inertia” measured
by parameter ρ. Notably, a dynamic structure like (3.11) is consistent with
recurrent empirical estimates of IS equations, which almost invariably find
both (a) and (b) - two features that are not easily accommodated in the
Let Ŷt = Ztm and Ŷt+1 = Ztn . Then it is possible to write Ŷt+1 = Ytρ Ztα for linear
combinations of the parameters ρ and α, such that ρm + α = n.
16
16
framework of the NNS17 . We shall see that this specification entails considerable differences also in the dynamic properties of the economy. Let us now
turn to the AS function (3.10) in its straightforward log-version:
π̂t+1 = β ŷt+1
(3.12)
Again there are analogies, but also important differences in comparison with
the NNS model. The function describes the price/output dynamics off the
a
being the deviation
AS curve that is associated with the IGE, with β = 1−a
of current from expected inflation that is necessary for competitive firms to
supply one unit of profit-maximizing output above/below potential. Consequently, the key to the inflation gap is the difference between the actual
rate and its expected value ex ante rather than the rate expected for the
future. The flavour of (3.12) may in that sense be more Lucasian than New
Keynesian, but it captures the essence of Wicksell’s idea of inflation as an
expectational disequilibrium phenomenon combined with Keynes’s idea of
the labour market being affected by the unwarranted coincidence between
nominal wage contracts and their real value. It is clear that at this point
a consistent model calls for additional hypotheses about expectation formation and interest-rate determination. Before proceeding to these two steps,
it is useful to check the dynamic properties of the two-equation system
(3.11)-(3.12).
3.4
A model check
Equations (3.11) and (3.12) form a first-order difference system in the two
gaps [ŷt+1 , π̂t+1 ] with exogenous nominal interest rate and expected inflation. This formulation is sufficient for a preliminary check of its dynamic
properties, compared with Wicksell’s theory of cumulative processes of inflation and Keynes’s restatement of that theory in terms of output.
To begin with, it is convenient to define the variable i ≡ r∗ + π ∗ in the IS
function (3.11), the nominal value of the natural interest rate. It corresponds
to Wicksell’s ”normal rate of interest” (Wicksell, 1898, p. 82), in modern
parlance: the NAIRI, which provides the IGE benchmark for the nominal
interest rate it . Let ît = it − i∗ be the nominal interest-rate gap (exactly
equivalent to the real gap). We thus have the following non-homogeneous
system:
ŷt+1
ρ 0
ŷt
−α
=
+
[ît ]
π̂t+1
βρ 0
π̂t
−βα
17
Attempts to fix the problem usually amount to injecting additional ”frictions” into
the markets, or postulate limits to the information-processing capacity of agents. Examples of inertial market frictions can be found in Woodford (2003b, ch.5) and Aghion
et al. (2004). Informational imperfections have been investigated by Mankiw and Reis
(2003), Sims (2003), Orphanides and Williams (2006). The consideration of the savinginvestment imbalances, as in the original Wicksellian macroeconomics, may be seen as a
more straightforward approach to serial correlation.
17
For any initial value î0 6= 0, this possesses the following steady-state solutions:
α
ŷ = −
î0
(3.13)
1−ρ
π̂ = −
βα
î0
1−ρ
(3.14)
That is to say:
Proposition 3 A permanent interest-rate gap determines permanent output and inflation gaps. Conversely, the output and inflation gaps are nil
only if the interest-rate gap is also nil (see also Leijonhufvud, 1981, p.136).
Proposition 4 If ρ ∈ [0, 1] output and inflation converge monotonically to,
and remain locked in, the values given by (3.13) and (3.14), with both output
and inflation being inefficiently high or low, and being inconsistent with their
IGE expected values.
These two propositions capture the essence of WK cumultative processes as
disequilibrium phenomena.
Consider the typical Wicksellian case where î0 < 0 and the intial inflation is zero so that π ∗ = 0. Consequently, the price level is set on the
path given by (3.14), growing indefinitely at a constant rate. In the Wicksellian literature, cumulative processes are, however, often associated with
non-monotonic, accelerating inflation rates18 . Our assumption that inflation
expectations are held constant at π ∗ = 0 corresponds to a stage that Wicksell describes as a relatively favourable situation where expectations remain
anchored to the ”normal” price level, and changes are deemed temporary.
However, as long as the interest-rate gap is not closed, changes in the price
level persist. This raises the questions of how expectations are revised, and
how the revision mechanism impinges upon the dynamic process - problems
that will be reconsidered below.
Our model shows that a cumulative process unfolds on the real side of the
economy, too. This illustrates the Keynesian point that saving-investment
imbalances are misallocations that require real resource adjustments, irrespective of the degree of flexibility of prices. In a typical Keynesian situation
with î0 > 0, resulting from a fall in the ”marginal efficiency of capital”, equation (3.13) indicates that output will converge to a steady state characterized
by a permanent gap with respect to the IGE. At that point of effective demand, the goods market clears, while the output gap implies unemployment
above the natural rate (if there is any).
18
Wicksell (1922, p. XII n.1) explained this as part of the mechanism of expectations
formation: ”[A]s long as the change in prices . . . is believed to be temporary, it will in fact
remain permanent; as soon as it is considered to be permanent, it will become progressive,
and when it is eventually seen progressive it will turn into an avalanche”.
18
It might be argued that our model is not fully true to Wicksell, who envisaged the cumulative process as a mechanism of price adjustments only, nor
to Keynes, who emphasized that aggregate demand and aggregate supply
could contract in a multiplier process at an unchanged price level. However,
these mutually exclusive views oversimplify the processes driven by savinginvestment imbalances in a flex-price economy, as shown by our model; and
they oversimplify the positions of the two original authors.Wicksell (1898, p.
142-3; 1915, p. 1955) did not deny that cumulative inflation is accompanied
by changes in output; he just considered them to be non-cumulative and,
hence, less relevant19 . Nor did Keynes exclude that the price level changes
when the real-income mechanism is at work (e.g.,1936, ch. 19). He only
pointed out that downward price (and wage) flexibility would tend to make
things worse and that, on the other hand, price level stability would not be
a sufficient condition for full-employment equilibrium, at which the money
rate of interest would accord with the natural rate, or ”neutral rate” in
Keyness diction (1936, p. 243).
As a synthesis of Wicksellian and Keynesian ideas the essence of our
model is to demonstrate that booms and slumps as well as inflation and
deflation, are intertemporal disequilibrium phenomena in three distinct, but
interconnected, meanings: (i) excess investment or saving is accommodated
at the ”wrong” real interest rate, (ii) the goods market clears at the ”wrong”
levels of output and inflation, and (iii) the expected rate of inflation is
”wrong” with respect to actual inflation.
4
Expectations and stabilization in the WicksellKeynes Triangle
Our basic WK model warrants further exploration with regard to the formation of expectations and its consequences for monetary policy. In the basic
version we assumed exogenous expectations that reflect the ”normal” growth
rate (zero or positive) of the price level. The problem with this assumption is
that, in the case of a (persistent) interest-rate gap, expectations of return to
normality will be systematically falsified. While modern economists tend to
rule this out by focusing exclusively on states of the economy in which expectations are statistically correct, Wicksell, Keynes and many others in their
19
Wicksell’s theory of the cumulative process is often interpreted in terms of the traditional neoclassical view that perfectly competitive markets would keep resources fully
employed to the extent that output cannot be increased and a low market interest rate
could only generate inflation. However, the key to such processes is that firms are allowed
to over-borrow and demand additional capital. It is only an extreme idea of full employment that may preclude this demand from being met by mobilizing or relocating resources
within the economy. This was a major objection that Lindahl, Myrdal and Hayek raised
against Wicksell from a multi-sector general equilibrium point of view (see Boianovsky
and Trautwein, 2006; 2006a).
19
circles were concerned with tracking the economy’s behaviour outside expectational equilibrium. Nevertheless, continuous under- or over-estimation of
inflation was deemed an untenable assumption even in their times. Wicksell
(1915, p. 196; 1922, p. XII) therefore introduced the hypothesis of learning
in the cumulative process that shifts expectations from static to adaptive
to forward-looking and eventually ”rational” in the sense of self-fulfilling20 .
As we cannot go deeper into the literature here, we confine ourselves to
examining a noteworthy coincidence between the expectation mechanisms
considered in the Swedish school and the way in which rational expectations
are introduced in the standard NNS model.
4.1
The troublesome role of rational expectations
An analogy between the Swedish school and modern macroeconomics can be
stated in terms of short-run rational expectations, the (statistically correct)
e −π
anticipation in period t of the one-period inflation rate, Et (πt+1
t+1 ) = 0,
e
which implies πt+1 = Et πt+1 . Even though we skip the learning process, we
can endogenize expectations in a way that Wicksell regarded as plausible,
but also as worrysome. Unlike the NNS, he was not concerned with jumps
from one equilibrium to the next, but with the convergence of expectations
in a disequilibrium process. Therefore, it is useful to re-examine the WK
model in the case of an economy where a fraction δ of the agents holds
short-run rational expectations of inflation, Et πt+1 , whereas a fraction 1 − δ
sticks to the expectation of ”normal” inflation, π ∗ . To this effect, we replace
π ∗ in equations (3.11) and (3.12) with δπt+1 + (1 − δ)π ∗ , while maintaining
that π̂t+1 = πt+1 − π ∗ . As a result,
ŷt+1 = ρ0 ŷt − α0 ît
(4.1)
π̂t+1 = β 0 ŷt+1
(4.2)
where
α0 = α
1−δ
1−δ
β
, ρ0 = ρ
, β0 =
1 − δ(1 + αβ)
1 − δ(1 + αβ)
1−δ
The steady-state solution for [ŷt+1 , π̂t+1 ] can simply be restated as follows
ŷ = −
α0
î0
1 − ρ0
(4.3)
π̂ = −
β 0 α0
î0
1 − ρ0
(4.4)
20
Lundberg (1930) actually used the notion of ”rational expectations”. Lindahl (1930,
p. 147) had it in the sense of ”individual anticipations of coming price developments”
that are ”the causes of the actual developments themselves”. For a modern treatment of
the learning process in the cumulative process see Howitt (1992).
20
Though similar to (3.11) and (3.12), these new solutions are ambiguous
with regard to their sign, magnitude and stability, essentially in connection
with parameter δ. In general, we find that:
i the coefficients of ŷ and π̂ increase with δ in absolute value; forwardlooking expectations amplify the deviation of the steady state from
the IGE path,
ii for ŷ and π̂ to maintain the normal negative relationship with î0 , δ
−1
αβ
should be bounded at δ < 1 + 1−ρ
< 1; a higher share of forwardlooking expectations would invert the relationship between interestrate gap, output gap and inflation gap (e.g., a positive interest-rate
gap would raise inflation permanently),
iii if δ satisfies the sign condition, the system also converges monotonically to [ŷ, π̂]; if δ exceeds the sign condition, the system may take
different trajectories (all with positive sign with respect to î0 ) some of
which may be explosive,
iv the limit solution for δ → 1, is [ŷ, π̂] = [0, î0 ]; in this case, the system
”jumps” to an inflation gap equal to (and of the same sign of) the
interest-rate gap, and forward-looking expectations are (self-)fulfilled.
These results illustrate the troublesome role of ”smart” short-run expectations in cumulative processes in the traditions of Wicksell and Keynes.
The problems arise in the IS function. Suppose again that î0 < 0,, with the
negative interest rate gap producing a positive output gap. As some agents
anticipate higher inflation, the market real interest rate is reduced further,
increasing the gaps, and so on. This ”expectation multiplier” explains why
short-run rational expectations are deviation-amplifying and why the cumulative process is bounded only if their weight is limited.
Recalling the discussion in section [3.1], it is worth stressing that the case
of δ → 1 replicates the result of McCallum (1986), according to which any
pegging of the nominal interest rate above or below the natural rate leads
to price-level changes that carry the sign opposite of what Wicksell predicted. McCallum stressed that this is consistent with the Fisher equation
and concluded that Wicksell’s theory does not hold under rational expectations. However, starting from the Fisher equation as a basis for expectation
formation, as McCallum does, is not a correct rendition of Wicksell’s theory,
in which the Fisher equation should be the end point of adjustments in a
disequilibrium process. As can be seen from our treatment, McCallum’s conclusion is valid only within the limits of uniformly held rational expectations
(see also Howitt, 1992).
21
4.2
Endogenizing the interest rate and anchoring expectations
In view of the processes driven by saving-investment imbalances, Wicksell
and Keynes raised two crucial consequential questions: how can interestrate gaps be closed? Can capital market forces take care of the problem or
is a ”visible hand” required? It is worth comparing answers from the WK
triangle with those in the NNS triangle.
Wicksell was aware that, in the context of his theory, price-level stability
would require two conditions to be met: the nominal interest rate must be
connected to changes in the price level in a stabilizing way, and inflation expectations must be anchored by a norm against which price movements can
be gauged. Wicksell (1898, ch. 11) interpreted the natural rate of interest,
not as a variable that can be observed by anyone in the system, but as a
hidden attractor of the system, where the latter is driven by agents reacting
to observable market signals21 . Explaining the observable co-movement of
prices and interest rates22 , as a first approximation, Wicksell (1898, p. 11318) argued that cumulative processes might be self -correcting, provided that
commercial banks react to accelerating inflation by adjusting their lending
rates to a level consistent with the (new) steady-state level of prices. This,
however, requires that their expectations are anchored to a ”normal” rate of
inflation23 . Technically speaking, with reference to our WK model, Wicksell’s first move consists of endogenizing the nominal interest rate, and hence
the interest-rate gap. This operation transforms the non-homogenous system (4.1)-(4.2) into a homogeneous one, where all three gaps appear as
endogenous variables. In general, one expects homogenous systems to have
zero-gap solutions in the final steady state, which we are looking for. These
solutions should include the ”normal” inflation rate. In modern terminology, the aim is to define an ”interest rate rule” that supports a determinate
rational-expectations equilibrium. To address this issue, we may begin with
a simple representation of an indexation mechanism, such as the following:
e
it+1 = it + γ(πt+1 − πt+1
)
(4.5)
Starting from it , the interest rate will remain constant as long as inflation is in line with the expected rate, whereas it will increase (decrease)
as inflation accelerates (decelerates). In a Wicksellian perspective, equation
(4.5) can be interpreted as shorthand either for the price-level correlation of
the interest rate in the market process, or for the policy prescription that
can be found in Wicksell’s following formulation of a simple interest-rate
rule:
21
We owe this point to Axel Leijonhufvud.
This phenomenon was later described as ”Gibson paradox” by Keynes (1930, p. II:
198-208).
23
Note that, apart from the central bank, there are no forces in the NNS model that
would make the interest rate change with inflation.
22
22
”So long as prices remain unaltered, the [central] banks rate
of interest is to remain unaltered. If prices rise, the interest rate
is to be raised, and if prices fall, the rate of interest is to be
lowered; and the rate of interest is henceforth to be maintained
at the new level, until a further movement of prices calls for a
change in one direction or the other”. (Wicksell, 1898a, p. 102)
The key difference between such a strategy of monetary policy and marketbased interest-rate determination can only lie in the specification of inflation expectations. It is obvious that the interest-rate mechanism of equation
(4.5) is not sufficient to obtain the homogenous transformation as long as
e
πt+1
6= π ∗ . Yet this is precisely the problem with unregulated market processes examined in the previous paragraph. Hence, there is a clear role for
the coordination of inflation expectations through an interest-rate rule for
monetary policy. This can be derived as follows:
it+1 = it + γ(πt+1 − π ∗ )
(4.6)
Subtracting the (constant) NAIRI i∗ from both sides of equation (4.6), we
have
ît+1 = ît + γ π̂t+1
(4.7)
Equations (4.1), (4.2) and (4.7) form the homogeneous system in the three
gaps [ŷt+1 ; π̂t+1 ; ît+1 ], which we are looking for:


 

ŷt+1
ŷt
ρ0
0
−α0
 π̂t+1  =  ρ0 β 0 0
−α0 β 0   π̂t 
γβ 0 ρ0 0 1 − γβ 0 α0
ît+1
ît
Clearly, this system admits of a zero-gap steady-state solution. Thus our
model demonstrates a point argued by Wicksell (1898, 1915) and even more
forcefully by Lindahl (1930): the central bank must announce an inflation
norm, say π ∗ , and then gear the interest rate appropriately, in order to
provide an anchor for long-run rational expectations, by which the system
is stabilized.
Equation (4.7) can be regarded as a prototype of modern inflation targeting that we may call the Wicksellian rule, and it accords well with the NNS
view of the central bank as ”manager of expectations” (Woodford, 2003b, p.
15). However, the theoretical foundations of this consensus are substantially
different and, as will be shown in the following, they yield strongly different
conclusions for the design of monetary policy .
It might be objected that Keynes (1936) put forward an alternative explanation of interest-rate gaps (liquidity preference), an alternative mechanism of endogenous interest-rate adjustments (real balance effects) and an
alternative view of monetary policy (quantity control of money supply). He
also attached more importance to bond price expectations than to goods
23
price expectations. These differences are important, as real balance effects
and quantitative control came to predominate in the theory of monetary
policy throughout most of the 20th century. However, seen from the vantage point of the WK triangle, there are substantial analogies as Keynes,
like Wicksell, concluded that interest-rate gaps should not be left to unfettered market forces. He only used different ingredients to pursue the same
”modelling strategy” as Wicksell, namely endogenizing the interest rate by
way of monetary policy in order to realign the nominal interest rate to the
rate that ensures saving-investment equilibrium at full employment24 . Since
our purpose here is to compare the WK model with the NNS model, we now
concentrate on our Wicksellian interest-rate rule.
4.3
Exploring the Wicksellian rule
An endogenous interest-rate is a necessary, but not sufficient, condition for
convergence to the zero-gap state. It is convenient to note that previous
system can be reduced to the variables [ŷt+1 ; ît+1 ] since the path of π̂t+1 is
fully determined by them. With this specification, the system presents two
eigenvalues; convergence and stability require them to lie within the unit
circle. Next, a simple boundedness condition on γ is obtained.
Proposition 5 An endogenous interest rate indexed to the inflation-rate
gap ensures convergence and stability with respect to the zero-gap steady
state provided that the relevant inflation parameter is bounded within
0<γ<
2(1 + ρ0 )
α0 β
This is obtained by applying the Schur Principle to the coefficient matrix of
the relevant specification of the system25 . It has some interesting implications.
The first implication is that the Wicksellian rule need not - and should
not - specify reactions to both output and inflation gaps, since the two are
positively correlated. Stabilizing inflation also stabilizes output, and vice
versa. As we shall see, the crucial concern for monetary policy is not a
trade-off between inflation and output, but the speed and amplitude of the
adjustment process driven by the interest-rate policy.
The second implication of Proposition 5 is a contrast between the boundedness on γ and the ”Taylor principle”, which prescribes an inflation coefficient with a lower bound equal to 1 (see, e.g., Woodford, 2003b, p. 253-54).
The upper bound on γ is the more binding, the more inflation is sensitive
24
As shown by Tamborini (2006), there are in fact remarkable analogies between our
system and a system in which the nominal interest rate is governed by a Keynesian LM
function.
25
If A is the matrix, then, 1+tr(A)+det(A) > 0, 1−tr(A)+det(A) > 0, 1−det(A) > 0.
24
to interest-rate gaps (as measured by α0 β 0 ), and the more short-term inflation is anticipated (as measured by δ). The contrast between Proposition 5
and the Taylor principle arises from the differences in the microfoundations
underlying the IS relation. We have seen that, in the out-of-equilibrium
dynamics, interest-rate gaps affect present and future output and inflation
gaps. As a consequence, gentle, rather than aggressive, interest-rate corrections are required. A large coefficient may produce faster adjustment, but
tends to destabilize the system by overshooting reactions.
Our model thus suggests that compelling requirements of convergence
and stability are overlooked in much of the current literature on rules for
monetary policy. Special attention should be paid to parameters γ and δ,
because of the deviation-amplifying role that short-run expectations play in
the WK model. To appreciate this, it should be noted that the stability
region of Proposition
5 features two different regimes: monotonic conver√
(1− ρ0 )2
gence for γ < α0 β 0 , and damped oscillations otherwise. Consider the
following example, where a = 0.4 and ρ = 0.3 are taken as primitive (realistic) values, which generate α = 0.17 and β = 0.67. Choosing γ = 1 as
benchmark value, we would have monotonic convergence, if δ = 0, whereas
δ = 0.7 would generate damped oscillations26 . As δ approaches 1, the γ
upper bound approaches 0 and then becomes negative. That is to say, the
system oscillates and becomes unstable even for small values of γ (unless the
sign of the rule is inverted, lowering the interest rate when inflation is low,
and vice versa). The search for inbuilt oscillations engaged early mathematizers of Wicksellian and Keynesian ideas under the presumption that business cycle theory ought to be able to reproduce cycles endogenously. That
requirement has been dropped with the advent of modern DSGE methodology, which contents itself with ad-hoc calibration of stochastic processes of
exogenous shocks.
Finally, it should be noted that the Wicksellian rule is ”robust” in the
sense of Orphanides and Williams (2002). Our model shows that, contrary
to standard formulations of Taylor rules in the NNS literature, stabilization
through the Wicksellian rule does not require direct information about the
natural rates of interest or output. This is essential, as the unavailability
of such information is a key hypothesis in the Wicksell-Keynes triangle.
Our results also accord well with a growing literature that questions the
hypothesis of timely and precise information about the natural rates in the
NNS27 . Like the ”difference rules” discussed by Orphanides and Williams
(2006), the Wicksellian rule of equation (4.7) belongs to the class of adaptive
rules that, using step-by-step adjustments in view of observable changes in
the economy, may drive the latter back to intertemporal equilibrium.
26
For δ = 0, the system starts oscillating for γ > 1.83, becoming unstable for γ > 23.3.
For δ = 0.7, the system starts oscillating for γ > 0.23 and becomes unstable for γ > 5.6.
27
See, e.g., Orphanides and Williams (2002; 2002), Primiceri (2006) and Tamborini
(2008).
25
5
Conclusions
Answering the question, what Wicksell and Keynes knew about macroeconomics that modern economists fail to consider, we may now say ”a
lot”. Attempting to capture some of their central insights our model shows
that the macroeconomics of saving-investment imbalances, which dominated
business-cycle theory in the first half of the 20th century, is not just a collection of pre-scientific insights. Its essential ingredients are amenable to
rigorous treatment according to modern standards, with minimal deviations
from the standard perfect competition model. Essentially all that is required is that the natural rate of interest should be volatile, and that it
should not be easily transmitted to the capital market. Recent history testifies that these two conditions are sufficiently common and recurrent as
to make saving-investment imbalances worth of being brought back to the
forefront.
Contrasting the NNS triangle of intertemporal optimization, imperfect
competition and sticky prices with the WK model - defined by the triangle
of intertemporal coordination, imperfect capital markets and wrong interest
rates -, we have found that, even though the two structures can be made to
look similar, they lead to strongly different conclusions about the dynamic
properties of systems in which interest-rate gaps occur. We believe that our
conclusions provide an interesting basis for extending the scope of analysis
of business cycles and of the role of monetary policy. The two triangles
may converge towards the view that interest-rate rules are a crucial means
to ensure stability. Yet it is obvious that the design of such rules requires
further exploration, once it is accepted that the central bank may be misinformed about the natural interest rate, and that the dynamic properties of
processes with saving-investment imbalances differ from those of the DSGE
models (see e.g. Tamborini, 2008).
A critical issue, highlighted by our WK model through the interaction
of parameters δ and γ, is the distribution of ”short-term” and ”long-term”
inflation expectations and its possible endogenous changes - an interaction
that gives rise to rich and complex dynamic processes. Another issue to
be addressed is the effect that saving-investment imbalances may have on
the capital stock and, hence, the further evolution of the economy. Once
the additional capital is in place (or once a part of the previously optimal
capital stock is scrapped), the economy can actually produce more (or less),
slowing down (pushing up) inflation - regardless of whether this was the
result of a wrong interest-rate signal or not. This phenomenon, which can
be roughly described as aggregate demand and aggregate supply ”moving
together” (Greenwald and Stiglitz, 1987), is only marginally touched upon
in the NNS literature (e.g. in Wooford, 2003b, ch. 5). Further extensions of
the WK model should involve a more explicit treatment of financial structures, asset prices and their relations to the concept of the natural rate of
26
interest. Given the analytical restrictions of the NNS triangle in all these
respects, the scope of modelling in the spirit of Wicksell and Keynes merits
further exploration.
References
Aghion, P., Frydman, R., Stiglitz, J. E., Woodford, M., 2004. Information
and Expectations in Modern Macroeconomic: in Honor of Edmund S.
Phelps. Princeton University Press, Princeton.
Bernanke, B., Gertler, M., 1990. Financial Fragility and Economic Performance. Quarterly Journal of Economics 105, 87–114.
Blanchard, O., 1997. Comment on Goodfriend and King. NBER Macroeconomics Annual 12, 289–293.
Blanchard, O., 2000. What do we know about Macroeconomics that Fisher
and Wicksell did not? Quarterly Journal of Economics 115, 1375–1409.
Blanchard, O., 2008. The State of Macroeconomics. NBER Working Paper
14259.
Blanchard, O., Kiyotaki, N., 1987. Monopolistic Competition and the Effect
of Aggregate Demand. American Economic Review 77, 647–666.
Boianovsky, M., Trautwein, H. M., 2006. Wicksell after Woodford. Journal
of History of Economic Thought 28 (2), 171–185.
Boianovsky, M., Trautwein, H. M., 2006a. Price Expectations, Capital Accumulation and Employment: Lindahl’s Macroeconomics drom the 1920s
to the 1950s. Cambridge Journal of Economics 30, 881–900.
Borio, C., Lowe, P., 2002. Asset Prices, Financial and Monetary Stability:
Exploring the Nexus. Bank of International Settlements 114.
Casares, M., McCallum, B. T., 2000. An Optimizing IS-LM Framework
With Endogenous Investment. NBER Working Paper 7908.
Cassel, G., 1918. Theoretische Sozialökonomie. C.F. Winter, Leipzig.
Cooper, R., 2004. Monopolistic Competition and Macroeconomics: Theory
and Quantitative Implications. In: Brakman, S., Heijdra, B. J. (Eds.), The
Monopolistic Competition Revolution in Retrospect. Cambridge University
Press, Cambridge.
Dixit, A., Stiglitz, J. E., 1977. Monopolistic Competition and the Optimum
Product Diversity. American Economic Review 67, 297–308.
27
Greenwald, B. C., Stiglitz, J. E., 1987. Keynesian, New Keynesian and New
Classical Economics. Oxford Economic Papers 39, 119–133.
Hayek, F. A., 1931. Prices and Production. P.S. King, London.
Howitt, P., 1992. Interest Rate Control and Nonconvergence to Rational
Expectations. Journal of Political Economy 100, 776–800.
Keynes, J. M., 1930. A Treatise on Money. Vol. 1 - The Pure Theory of
Money. MacMillan.
Keynes, J. M., 1936. The General Theory of Employment, Interest and
Money. MacMillan, London.
Laidler, D. W., 2006. Woodford and Wicksell on Interest and Prices. The
Place of the Pure Credit Economy in the Theory of Monetary Policy.
Journal of History of Economic Thought 28, 151–60.
Leijonhufvud, A., 1981. The Wicksell Connection: Variation on a Theme. In:
Information and Coordination. Essays in Macroeconomic Theory. Oxford
University Press, New York.
Leijonhufvud, A., 2008. Keynes and the Crises. CEPR Policy Insight 23.
Lindahl, E. R., 1930. Penningpolitikens medel. Förlagsaktiebolaget, Malmö.
Lundberg, E., 1930. Om begreppet ekonomisk jämvikt. Ekonomisk Tidskrift
32, 133–160.
Lundberg, E., 1937. Studies in the Theory of Economic Expansion. P.S.
King, London.
Mankiw, G. N., Reis, R., 2003. Sticky Information: A Model of Monetary
Nonneutrality. In: Aghion, P., Frydman, R., Stiglitz, J. E., Woodford,
M. (Eds.), Information and Expectations in Modern Macroeconomic: in
Honor of Edmund S. Phelps. Princeton University Press, Princeton.
McCallum, B. T., 1986. Some Issues Concerning Interest Rate Pegging, Price
Level Indeterminacy, and the Real Bills Doctrine. Journal of Monetary
Economics 17, 135–160.
Modigliani, F., 1944. Liquidity Preference and the Theory of Interest Rate
and Money. Econometrica 12, 45–88.
Myrdal, G., 1931. Om penningteoretisk jämvikt. En studie över den ”normala räntan” i Wicksells penninglära. Ekonomisk Tidskrift 33, 191–302.
Orphanides, A., Williams, J. C., 2002. Robust Monetary Policy Rules with
Unknown Natural Rates. Brookings Papers on Economic Activity 2, 63–
118.
28
Orphanides, A., Williams, J. C., 2006. Inflation Targeting Under Imperfect
Knowledge. CEPR Discussion Paper Series 5664.
Patinkin, D., 1989. Money, Interest, and Prices : An Integration of Monetary and Value Theory. Cambridge University Press, Mass.
Pigou, A. C., 1933. The Theory of Unemployment. MacMillan, London.
Primiceri, G., 2006. Why Inflation Rose and Fell: Policy-Makers Beliefs and
U.S. Postwar Stabilization Policy. Quarterly Journal of Economics 121,
867–901.
Sims, C., 2003. Implications of Rational Inattention. Journal of Monetary
Economics 50, 665–690.
Tamborini, R., 2006. Back to Wicksell? In search of the foundations of practical monetary policy. Working Paper, Università degli Studi di Trento.
Tamborini, R., 2008. Monetary Policy with Investment-Saving Imbalances.
Working Paper, Università degli Studi di Trento.
Wicksell, K., 1898. Geldzins und Gueterpreise. Eine Untersuchung uber
die den Tauschwert des Geldes bestimmenden Ursachen. Gustav Fischer,
Jena.
Wicksell, K., 1898a. The Influence of the Rate of Interest on Commodity
Prices. In: Lindhal, E. R. (Ed.), Wicksell: Selected Papers in Economic
Theory. Allen and Unwin, London.
Wicksell, K., 1901. Föreläsningar i Nationalekonomi, I: Teoretisk Nationalekonomi. Berlingska Boktryckeriet, Lund.
Wicksell, K., 1915. Föreläsningar i Nationalekonomi, II: Om Penningar och
Kredit, 2nd Edition. Fritzes Hofbokhandel, Stockholm.
Wicksell, K., 1922. Vorlesungen über Nationalökonomie, Band 2: Geld und
Kredit. Gustav Fischer, Jena.
Woodford, M., 1999. Revolution and Evolution in Twentieth-Century
Macroeconomics. In: Gifford, P. (Ed.), Froniters of the Mind in the
Twenty-First Century. Harvard University Press.
Woodford, M., 2003a. Imperfect Common Knowledge and the Effects of
Monetary Policy. In: Aghion, P., Frydman, R., Stiglitz, J. E., Woodford,
M. (Eds.), Information and Expectations in Modern Macroeconomic: in
Honor of Edmund S. Phelps. Princeton University Press, Princeton.
Woodford, M., 2003b. Interest and Prices. Foundation of a theory of Monetary Policy. Princeton University Press.
29
Woodford, M., 2006. Comment on Symposium on Interest and Prices. Journal of History of Economic Thought 28, 186–198.
A
Appendix
A.1
Interest rate gaps and output gaps
Based on the model (3.1)-(3.8) in the text, we examine the allocations that
result if, starting in the steady state, the market real interest rate at time
t, Rt+1 , differs from the natural rate R∗ . Both rates are assumed to remain
constant thereafter.
To begin with, recall that the steady state is characterized by L∗ =
e
1 (fully employed labour force), πt+1
= πte = π ∗ (constant expected or
∗
”normal” inflation rate), Θ = R (households’ time discount factor equal to
the gross real return to capital, or natural rate of interest), C ∗ (constant
consumption), B ∗ = K ∗ (constant real stock of bonds representative of
capital stock), Y ∗ = H ∗ + R∗ K ∗ (households’ real income, given by labour,
H ∗ , and capital, R∗ K ∗ , incomes). Note that, as a consequence, St0 = It0 =
K ∗ , and once account is taken of capital replacement, net investment and
saving are equal and nil.
Turning now to a period t in which, cet. par., Rt+1 6= R∗ , let us first
examine households’ optimal consumption path (see equation (3.8)):
Ct+1 ∗
Ct = Et
(A.1)
R
Rt+1
Hence, with respect to the steady state, Rt+1 6= R∗ shifts consumption to
the present (if Rt+1 < R∗ ) or to the future (if Rt+1 > R∗ ). Parallelly,
households demand less or more (real) bonds, respectively.
Now let us see optimal investment of firms. This is (see equation (3.5)):
1
1−a
a
0
It = Kt+1 =
(A.2)
Rt+1
Hence, for Rt+1 6= R∗ investment is larger (if Rt+1 < R∗ ) or smaller (if
Rt+1 > R∗ ). Firms supply more (real) bonds in the forme case, less in the
latter.
As long as the central bank pegs Rt+1 6= R∗ , it should stand ready
to clear the excees supply of bonds (if Rt+1 < R∗ ) or the excess demand
of bonds (if Rt+1 > R∗ ), allowing households and firms to finance their
respective consumption and investment plans. However, these plans are not
mutually consistent in the goods market. In fact,
- if Rt+1 > R∗ , consumption is shifted from t to t + 1, while investment
in t, and the capital stock available in t + 1, are reduced. There is
excess supply in t and excess demand in t + 1,
30
- if Rt+1 < R∗ , the excesses are reversed.
How can these inconsistent plans be transformed into mutually consistent
demand and supply of output? To address this point, we follow the same
procedure as in the NNS model, plugging the budget constraint period by
period(see equation (3.7)) into households’ Euler equation:
Ht+1 + Rt+1 Bt+1 − Bt+2 ∗
∗ ∗
R
(A.3)
Ht + R K − Bt+1 = Et
Rt+1
The saving-investment inconsistency leads to Bt+s 6= Kt+s for s = 1, . . . ,
where the real value of the stock of bonds purchased by households differs
from the actual stock of capital goods purchased by firms at each point in
time. This results in wrong resource accounting. As long as Rt+1 6= R∗ , the
actual consumption path consistent with Bt+s = Kt+s should satisfy:
Yt+1 − Kt+2 ∗
(Yt − Kt+1 ) =
(A.4)
R
Rt+1
where Yt = Ht + R∗ K ∗ and Yt+1 = Ht+1 + Rt+1 Kt+1 . This reformulation of
households’ consumption path leads to the following propositions:
1. Given the capital stock chosen by firms for Rt+1 6= R∗ , there exists
a unique intertemporal vector of output realizations associated with
consistent ex-post output market clearing.
2. These output realizations correspond to non-zero gaps with respect to
the level of ”potential output” given by the capital stock that would
obtain with the natural rate of interest R∗ .
The proof goes as follows. First, for Rt+1 constant, Kt+2 = Kt+1 . Hence,
(A.4) can be rewritten as:
Yt+1 ∗
R∗
Yt =
R + Kt+1 1 −
Rt+1
Rt+1
Now divide both sides by Y ∗ we will obtain the intertemporal relationship
between output gaps:
Yt
Yt+1 R∗
Kt+1
R∗
=
+
1−
Y∗
Y ∗ Rt+1
Y∗
Rt+1
Upon recollecting the following relationships:
1
a 1
1−a
a
1−a
a
∗
a∗
Yt+1 = Kt+1 , Y = K , Kt+1 =
, K∗ =
Rt+1
R∗
the two output gaps result:
Ŷt+1 =
R∗
Rt+1
31
a
1−a
(A.5)
Ŷt ≈
R∗
Rt+1
1
1−a
(A.6)
i
h
1
− R1∗
where the approximation concerns the multiplicative term 1 − a Rt+1
which, for sufficiently small rates, is close to 1. (A.5) and (A.6) show that
the main implication of the market real interest rate being set above (below) the natural rate is a sequence of negative (positive) output gaps each
∗
depending on the current interest-rate gap RRt+1 .
A.2
Inflation gaps
As to price determination in relation to output gaps, given the generalequilibrium real wage rate w∗ and capital stock K ∗ (see equation (3.5)),
potential output at any time t can also be expressed as:
∗
Y =K
∗a
1−a
w∗
1−a
a
(A.7)
Since the nominal wage rate for t be given by Wt = w∗ Pt−1 (1 + π ∗ ), firms
can adjust output for t by choosing the labour input upon observing the
t
current real wage rate wt = W
Pt . As a result,
Yt = K
∗a
1 − a 1 + πt
w∗ 1 + π ∗
1−a
a
(A.8)
Ceteris paribus, profit-maximizing firms are ready to expand (contract) output as long as πt , being greater (smaller) than π ∗ , increases (reduces) the
current nominal value of the marginal product of labour vis-á-vis Wt . Conversely, we can derive the Marshallian supply curve of firms, that is, the
1+πt
inflation gap Π̂t ≡ 1+π
∗ which supports a given output gap. Let me divide
(A.8) with (A.7):
a
Π̂t = Ŷt 1−a
32
(A.9)